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 The Latitudinal Pressure Gradient That Would Be Created On a Rotating Spherical Planet

Because of a planet rotation, the fluids on its surface would move to its equator unless a countervailing pressure gradient is established. Below is given a derivation of the pressure distribution on the surface of a rotating spherical planet which would be required to maintain surface equilibrium.

Let r be the distance to the center of the planet and Ω its rate of rotation. The latitude angle is denoted φ and the longitude θ At a latitude φ the distance to the axis of the planet is rcos(φ). The centrifugal force on a particle of mass dm at latitude φ and distance to the planet's center of r is

#### F = (dm)(rcos(φ))Ω²

This may be resolved into a radial and a latitudinal component; i.e.,

#### Fr = −Fcos(φ) = − (dm)(rcos²(φ))Ω² Fφ = Fsin(φ) = (dm)(rcos(φ)sin(φ))Ω²

The mass in an infinitesimal volume dV is ρdV where ρ is the fluid density. An infinitesimal volume between r and r+dr, φ and φ+dφ, and θ and θ+dθ is r²cos(φ)drdφθ. This can be represented as an area times a thickness in three ways:

## The Latitudinal Distribution of Pressure

The force on the infinitesimal element due the pressure p on the face at φ+dφ less the force on face at φ is

#### dA(∂p/∂φ)dφ

When this force is added to the latitudinal component of the centrifugal force on dV the result must be zero; i.e.,

#### −(∂p/dφ)dφdA + (ρdV)((rcos(φ)sin(φ))Ω²) = 0 which reduces to ∂p/∂φ = ρr²cos(φ)sin(φ)Ω² = ρr²(sin(2φ)/2)Ω²

Integrating this last equation with respect to φ from 0 to Φ with r constant gives:

#### p(Φ,r) - p(0,r) = ρr²Ω²[(cos(2Φ)-1)/4]

Therefore the equilibrium latitudinal pressure distribution is given by

## The Radial Distribution of Pressure

The area of the top and bottom surfaces of the infinitesimal volume are approximately equal to dB = r²cos(φ)dφθ. The infinitesimal volume is then dV=dBdr.

The difference in pressure on the top and bottom surfaces plus the gravitational pull on the infinitesimal volume plus the centrifugal force must equal zero; i.e.,

#### −(∂p/∂r)drdB −(ρdV)[g + rcos²(φ)Ω²] = 0 which reduces to (∂p/∂r) = −ρ[g + rcos²(φ)Ω²]

When this last equation is integrated with respect to r from R0 to R the result is:

#### p(φ,R) − p(φ,R0) = −ρg(R−R0) −ρ(cos²(φ)Ω²)(R²−R0²)/2

This result presumed the gravitational attraction is constant with altitude, R−R0. More generally the result would be the pressure as a function of altitude with a correction for the effect of the centrifugal force. The correction term depends upon the cosine squared of the latitude.